I have a question here regarding picking weapons. For example, weapon A has higher APS but lower physical damage, and vice versa with weapon B. Which one should I pick?

Nice necro. Anyways, the short answer is

It depends.

Seriously. It's dependent on the character build and play style. The beauty and challenge of PoE is the complexity of the game mechanics. For example, if I'm speccing elemental, I would probably want higher attack speed, as the proc effects roll per hit. On the other hand, if I'm doing a spectral throw build, I would probably want to go higher physical damage because the skill scales off your main weapon.

There are way too many possibilities to try to give you a comprehensive answer. I suggest reading up some more, visiting the poe wiki and just have fun and play the game. That's the best way to do it.

Disagree with not specializing; that was the old logic. There are some damn fine specific-weapon nodes to grab since release.

Swords seem pretty awesome. The combined damage/life nodes in the Cruel Blade, Razor's Edge, and Slashing Comeback clusters are all very appealing, as are the damage/armour nodes in the Blade of Renown cluster.

The only way I'd imagine going axes is if I was in the Templar area (to grab Fell the Weak) and somehow wasn't going down towards the Duelist area of the tree. I don't really believe that those nodes alone would compel me to specialize, and there are no damage/life combo nodes for axes.

Hell, I'd consider maces more viable than axes. Baptism by Light ties with Fell the Weak, and the Smashing Blows cluster at least gives it some damage/life hybrid. And, as mentioned earlier, claws are pretty awesome, as are daggers.

Also considering that there are two types of swords (slashing and thrusting), which means specializing in them still leaves options open, and I'd enthusiastically say swords, assuming axes are the only competition.

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In general, I've found that RT builds favor axes over swords. Not just because of the sword accuracy bonus, but also if you're up north to grab RT you're likely to grab some nice axe notables like Fell the Weak and Runed Cleaver.